Obviously, this ball has a spherical symmetry and could be considered as a "smooth" distribution (under the limit of an infinite set of points).

Now, I would like to randomly "erode" the distribution, to get something that look a bit like a fractal shape (think about how an eroded mountain could be obtain from a smooth hill). How can I do that ? What Mathematica procedures could modify the Ball distribution that act like a random eroding process ?

To me, "eroding" mean removing a randomly selected point, and several of its neighbors, then repeat the process several times.

(1) I would use RandomReal instead of Random. (2) Not really clear how "erode" is intended to operate. You might want to elaborate on that. Is the idea to pick a point at random and remove it along with several of its closest neighbors?
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Daniel LichtblauFeb 6 '14 at 14:47

Eroding mean randomly removing points, yes. How can I ask Mma to remove a random point and several of its neihbors, and do it again for many other randomly selected points ?
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ChamFeb 6 '14 at 14:54

According to the eroding process I defined above, the final distribution may contain holes inside. This is an interesting possibility for what I'm trying to achieve. I'm also interested in an eroding process that act "from the exterior" only (no "holes" inside), but this is more ambiguous to me ; I don't know how to define the "surface" of a distribution of points. Select first a point which is farthest from the center (origin) ?
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ChamFeb 6 '14 at 15:06

By using the function FindClusters with the Method -> "Agglomerate" you can split the points into clusters (I chose to split into 500 clusters, but you can tune this to your preferences). After a sort which puts the largest cluster first (which will likely be the cluster of particles near the center) you can "erode" by included the first n components you desire.

Eroding from the outer regions first is slightly trickier. You might first invert all points e.g. dividing by distance squared from origin, so that the furthest are now near the center. Randomly choose from those, and use erode in neighborhoods of the original values. For funkier effects, perhaps erode from the inverted set, so "neighbors" are not really neighbors in the Euclidean sense.

If you use ListPointPlot3D and ListSurfacePlot3d we get this cool animation. We highlight the points to be deleted during the current step with the red sphere denoting the active delete center.

Separately looking at the two cases now in the following first image we can animate the process for random delete in the cluster. Next one is the the animation when we delete points from the outer region of the cluster.

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